# Properties

 Label 845.2.c.h Level $845$ Weight $2$ Character orbit 845.c Analytic conductor $6.747$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 34 x^{16} + 407 x^{14} + 2175 x^{12} + 5555 x^{10} + 6664 x^{8} + 3544 x^{6} + 681 x^{4} + 47 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{6} ) q^{2} + ( 1 + \beta_{11} ) q^{3} + ( -2 - \beta_{8} + \beta_{9} - \beta_{10} ) q^{4} + \beta_{13} q^{5} + ( -\beta_{1} + \beta_{6} + \beta_{12} - \beta_{16} + \beta_{17} ) q^{6} + ( \beta_{12} - \beta_{13} - \beta_{17} ) q^{7} + ( 2 \beta_{1} - \beta_{6} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{8} + ( 1 + \beta_{3} - 2 \beta_{5} + \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{6} ) q^{2} + ( 1 + \beta_{11} ) q^{3} + ( -2 - \beta_{8} + \beta_{9} - \beta_{10} ) q^{4} + \beta_{13} q^{5} + ( -\beta_{1} + \beta_{6} + \beta_{12} - \beta_{16} + \beta_{17} ) q^{6} + ( \beta_{12} - \beta_{13} - \beta_{17} ) q^{7} + ( 2 \beta_{1} - \beta_{6} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{8} + ( 1 + \beta_{3} - 2 \beta_{5} + \beta_{9} + \beta_{11} ) q^{9} + ( -\beta_{2} + \beta_{10} ) q^{10} + ( \beta_{1} - \beta_{4} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{11} + ( -3 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{12} + ( \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} - \beta_{8} - 3 \beta_{10} ) q^{14} + ( \beta_{13} + \beta_{14} ) q^{15} + ( 3 - \beta_{3} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{16} + ( \beta_{2} - \beta_{5} + \beta_{7} - \beta_{11} ) q^{17} + ( -2 \beta_{4} + 3 \beta_{6} + 2 \beta_{12} - 2 \beta_{13} - \beta_{16} + \beta_{17} ) q^{18} + ( 2 \beta_{1} - \beta_{6} + \beta_{12} - \beta_{14} + \beta_{16} ) q^{19} + ( \beta_{4} + \beta_{6} - 2 \beta_{13} - \beta_{16} ) q^{20} + ( -\beta_{1} - 3 \beta_{6} + \beta_{12} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{21} + ( 3 - 2 \beta_{2} + 4 \beta_{5} + \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{22} + ( -2 + 2 \beta_{3} - \beta_{5} - \beta_{9} + \beta_{11} ) q^{23} + ( 2 \beta_{1} - \beta_{4} + 2 \beta_{6} - 3 \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{24} - q^{25} + ( 1 + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{27} + ( \beta_{1} + 2 \beta_{4} + \beta_{6} - 2 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} + \beta_{16} ) q^{28} + ( 3 + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} - \beta_{8} ) q^{29} + ( -\beta_{2} + \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{30} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{31} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - \beta_{12} - 2 \beta_{13} - 5 \beta_{15} + \beta_{16} - 3 \beta_{17} ) q^{32} + ( -2 \beta_{4} + 3 \beta_{6} + \beta_{12} + \beta_{14} - 2 \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{33} + ( -\beta_{4} - 2 \beta_{6} + \beta_{12} + 5 \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} ) q^{34} + ( 1 + \beta_{3} + \beta_{7} ) q^{35} + ( 2 + \beta_{2} - \beta_{3} + 7 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{36} + ( -\beta_{1} + 2 \beta_{4} + 3 \beta_{6} + \beta_{12} + 2 \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{37} + ( 6 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{38} + ( 2 \beta_{2} - \beta_{5} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{40} + ( -\beta_{1} + \beta_{4} - 3 \beta_{6} + \beta_{12} + 2 \beta_{13} + 6 \beta_{15} - \beta_{17} ) q^{41} + ( -3 + \beta_{3} - \beta_{5} + 3 \beta_{7} + 3 \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{42} + ( -4 + \beta_{2} + 2 \beta_{5} - \beta_{7} - \beta_{9} ) q^{43} + ( -3 \beta_{1} + 4 \beta_{4} + 6 \beta_{6} - 4 \beta_{12} - 6 \beta_{13} - \beta_{14} - 3 \beta_{15} - \beta_{17} ) q^{44} + ( -\beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{45} + ( \beta_{1} - \beta_{4} + 2 \beta_{6} - \beta_{13} - 3 \beta_{14} + 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{46} + ( 2 \beta_{6} - 2 \beta_{12} - 6 \beta_{13} + \beta_{14} - 3 \beta_{15} - \beta_{16} ) q^{47} + ( 6 - 4 \beta_{3} + 6 \beta_{5} - \beta_{8} + \beta_{9} + 9 \beta_{10} + \beta_{11} ) q^{48} + ( -3 - 2 \beta_{2} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{49} + ( \beta_{1} - \beta_{6} ) q^{50} + ( -5 + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{51} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{53} + ( -3 \beta_{4} + 5 \beta_{6} + \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{54} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{55} + ( 3 - 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{56} + ( 2 \beta_{1} - \beta_{4} - 3 \beta_{6} - 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{16} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{4} - 3 \beta_{6} + \beta_{12} + 7 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{58} + ( \beta_{1} - \beta_{6} - \beta_{12} + 2 \beta_{13} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{59} + ( -\beta_{1} + 3 \beta_{6} - \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{60} + ( -\beta_{3} - \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{61} + ( 7 - \beta_{3} + 8 \beta_{5} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{62} + ( -3 \beta_{1} - 4 \beta_{6} + \beta_{12} - \beta_{14} + 10 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{63} + ( 3 - \beta_{2} + 7 \beta_{5} + \beta_{7} - 6 \beta_{8} + 4 \beta_{10} + 3 \beta_{11} ) q^{64} + ( -2 - 3 \beta_{2} - \beta_{3} + 7 \beta_{5} - \beta_{7} - \beta_{9} + 5 \beta_{10} - 2 \beta_{11} ) q^{66} + ( \beta_{1} - \beta_{4} - 3 \beta_{6} + 2 \beta_{12} + \beta_{13} - 2 \beta_{15} + \beta_{17} ) q^{67} + ( -2 - 6 \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{68} + ( 1 - 3 \beta_{2} + 3 \beta_{3} - 5 \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{69} + ( -\beta_{1} + \beta_{4} + 3 \beta_{6} - \beta_{12} + 2 \beta_{15} + \beta_{17} ) q^{70} + ( \beta_{1} - \beta_{4} + 5 \beta_{6} - 3 \beta_{13} - \beta_{14} - 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{71} + ( -\beta_{1} + 3 \beta_{4} + 5 \beta_{6} - 3 \beta_{12} - \beta_{13} + 4 \beta_{14} - 3 \beta_{15} + \beta_{16} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{13} + 4 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{73} + ( -5 + 3 \beta_{2} + 4 \beta_{3} - \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - \beta_{9} - 5 \beta_{10} - \beta_{11} ) q^{74} + ( -1 - \beta_{11} ) q^{75} + ( -5 \beta_{1} + 3 \beta_{4} + 7 \beta_{6} - 3 \beta_{13} - \beta_{14} - 5 \beta_{15} + \beta_{16} - \beta_{17} ) q^{76} + ( -2 - 2 \beta_{2} + \beta_{3} - 4 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{77} + ( 3 + \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{79} + ( -2 \beta_{4} - \beta_{6} + \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{80} + ( 5 - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{81} + ( -5 - \beta_{2} - 5 \beta_{5} + 4 \beta_{8} + \beta_{9} - 7 \beta_{10} ) q^{82} + ( 2 \beta_{1} + \beta_{4} - 3 \beta_{6} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{83} + ( 4 \beta_{1} + 2 \beta_{4} - \beta_{6} - 5 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{84} + ( -\beta_{1} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{85} + ( 3 \beta_{1} - 6 \beta_{6} - \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{86} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{8} - 3 \beta_{10} ) q^{87} + ( -5 + 6 \beta_{2} - \beta_{3} - 4 \beta_{5} - 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{88} + ( -3 \beta_{4} - 2 \beta_{6} + 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} + \beta_{16} ) q^{89} + ( 2 + 2 \beta_{3} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{90} + ( 1 + \beta_{2} - 6 \beta_{3} + 4 \beta_{5} + 3 \beta_{7} + 5 \beta_{8} + 5 \beta_{10} ) q^{92} + ( \beta_{1} + \beta_{4} - 3 \beta_{6} + 5 \beta_{13} - \beta_{14} + \beta_{15} - 3 \beta_{16} + \beta_{17} ) q^{93} + ( 3 + 7 \beta_{2} + \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{11} ) q^{94} + ( 2 \beta_{2} + \beta_{3} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{95} + ( 4 \beta_{4} - 9 \beta_{6} + 3 \beta_{12} + 6 \beta_{13} + 3 \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{96} + ( 2 \beta_{1} + 2 \beta_{4} - 3 \beta_{6} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + 3 \beta_{16} ) q^{97} + ( \beta_{1} - \beta_{4} + 3 \beta_{6} - \beta_{12} - 12 \beta_{13} - 2 \beta_{14} - 8 \beta_{15} - 2 \beta_{16} - 4 \beta_{17} ) q^{98} + ( -3 \beta_{1} + 7 \beta_{6} - \beta_{12} - 2 \beta_{13} - 5 \beta_{15} - \beta_{16} + 5 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 14q^{3} - 34q^{4} + 32q^{9} + O(q^{10})$$ $$18q + 14q^{3} - 34q^{4} + 32q^{9} - 6q^{10} - 24q^{12} - 4q^{14} + 74q^{16} + 2q^{17} + 24q^{22} - 28q^{23} - 18q^{25} + 44q^{27} + 24q^{29} + 4q^{30} + 14q^{35} - 6q^{36} + 94q^{38} + 24q^{40} - 22q^{42} - 78q^{43} - 6q^{48} - 32q^{49} - 86q^{51} - 16q^{53} - 18q^{55} + 58q^{56} - 6q^{61} + 20q^{62} - 68q^{64} - 98q^{66} - 40q^{68} + 26q^{69} - 30q^{74} - 14q^{75} + 8q^{77} + 78q^{79} + 58q^{81} + 8q^{82} + 32q^{87} - 84q^{88} + 20q^{90} - 54q^{92} + 32q^{94} + 8q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 34 x^{16} + 407 x^{14} + 2175 x^{12} + 5555 x^{10} + 6664 x^{8} + 3544 x^{6} + 681 x^{4} + 47 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$314 \nu^{16} + 11531 \nu^{14} + 155943 \nu^{12} + 1000554 \nu^{10} + 3262234 \nu^{8} + 5215959 \nu^{6} + 3465895 \nu^{4} + 671259 \nu^{2} + 26257$$$$)/23348$$ $$\beta_{3}$$ $$=$$ $$($$$$30751 \nu^{16} + 1054474 \nu^{14} + 12808346 \nu^{12} + 70153600 \nu^{10} + 186147393 \nu^{8} + 235030088 \nu^{6} + 127580183 \nu^{4} + 19102030 \nu^{2} + 1292422$$$$)/957268$$ $$\beta_{4}$$ $$=$$ $$($$$$-35814 \nu^{17} - 1200088 \nu^{15} - 13998849 \nu^{13} - 71418339 \nu^{11} - 168476899 \nu^{9} - 178967853 \nu^{7} - 90666883 \nu^{5} - 26975687 \nu^{3} - 3665123 \nu$$$$)/478634$$ $$\beta_{5}$$ $$=$$ $$($$$$-85080 \nu^{16} - 2881455 \nu^{14} - 34266505 \nu^{12} - 181181422 \nu^{10} - 456103704 \nu^{8} - 541625697 \nu^{6} - 299565143 \nu^{4} - 67640277 \nu^{2} - 3980585$$$$)/957268$$ $$\beta_{6}$$ $$=$$ $$($$$$-335154 \nu^{17} - 11364485 \nu^{15} - 135353204 \nu^{13} - 716151604 \nu^{11} - 1791626870 \nu^{9} - 2047318863 \nu^{7} - 952755688 \nu^{5} - 100659691 \nu^{3} + 3349792 \nu$$$$)/957268$$ $$\beta_{7}$$ $$=$$ $$($$$$-363580 \nu^{16} - 12350593 \nu^{14} - 147564147 \nu^{12} - 785115096 \nu^{10} - 1982532264 \nu^{8} - 2298989213 \nu^{6} - 1088197267 \nu^{4} - 121505207 \nu^{2} - 1655871$$$$)/957268$$ $$\beta_{8}$$ $$=$$ $$($$$$-200967 \nu^{16} - 6775530 \nu^{14} - 79894886 \nu^{12} - 415449506 \nu^{10} - 1010742096 \nu^{8} - 1112574870 \nu^{6} - 521526326 \nu^{4} - 82079517 \nu^{2} - 3057070$$$$)/478634$$ $$\beta_{9}$$ $$=$$ $$($$$$435208 \nu^{16} + 14750769 \nu^{14} + 175561845 \nu^{12} + 927951774 \nu^{10} + 2319486062 \nu^{8} + 2656924919 \nu^{6} + 1269531033 \nu^{4} + 175456581 \nu^{2} + 8028849$$$$)/957268$$ $$\beta_{10}$$ $$=$$ $$($$$$491862 \nu^{16} + 16646116 \nu^{14} + 197617407 \nu^{12} + 1040169704 \nu^{10} + 2584684372 \nu^{8} + 2946880266 \nu^{6} + 1433654597 \nu^{4} + 222251342 \nu^{2} + 7961039$$$$)/957268$$ $$\beta_{11}$$ $$=$$ $$($$$$-584675 \nu^{16} - 19943268 \nu^{14} - 240060381 \nu^{12} - 1294904608 \nu^{10} - 3354455949 \nu^{8} - 4093410708 \nu^{6} - 2167738212 \nu^{4} - 368768660 \nu^{2} - 15114581$$$$)/957268$$ $$\beta_{12}$$ $$=$$ $$($$$$584675 \nu^{17} + 19943268 \nu^{15} + 240060381 \nu^{13} + 1294904608 \nu^{11} + 3354455949 \nu^{9} + 4093410708 \nu^{7} + 2167738212 \nu^{5} + 368768660 \nu^{3} + 15114581 \nu$$$$)/957268$$ $$\beta_{13}$$ $$=$$ $$($$$$-26257 \nu^{17} - 892424 \nu^{15} - 10675068 \nu^{13} - 56953032 \nu^{11} - 144857081 \nu^{9} - 171714414 \nu^{7} - 87838849 \nu^{5} - 14415122 \nu^{3} - 562820 \nu$$$$)/23348$$ $$\beta_{14}$$ $$=$$ $$($$$$1292422 \nu^{17} + 43911597 \nu^{15} + 524961280 \nu^{13} + 2798209504 \nu^{11} + 7109250610 \nu^{9} + 8426552815 \nu^{7} + 4345313480 \nu^{5} + 752559199 \nu^{3} + 41641804 \nu$$$$)/957268$$ $$\beta_{15}$$ $$=$$ $$($$$$1511745 \nu^{17} + 51340153 \nu^{15} + 613239633 \nu^{13} + 3263026086 \nu^{11} + 8258626383 \nu^{9} + 9697215893 \nu^{7} + 4870923842 \nu^{5} + 766476583 \nu^{3} + 32061737 \nu$$$$)/957268$$ $$\beta_{16}$$ $$=$$ $$($$$$973390 \nu^{17} + 33059157 \nu^{15} + 394965388 \nu^{13} + 2103186829 \nu^{11} + 5337102096 \nu^{9} + 6325494905 \nu^{7} + 3285808471 \nu^{5} + 584146866 \nu^{3} + 25690001 \nu$$$$)/478634$$ $$\beta_{17}$$ $$=$$ $$($$$$-3090823 \nu^{17} - 104771128 \nu^{15} - 1247295356 \nu^{13} - 6597016758 \nu^{11} - 16519804175 \nu^{9} - 19031863984 \nu^{7} - 9270352669 \nu^{5} - 1361362954 \nu^{3} - 48749924 \nu$$$$)/957268$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} + 2 \beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{17} - 3 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - \beta_{6} - 9 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{11} + 27 \beta_{10} - 9 \beta_{9} + 16 \beta_{8} + 5 \beta_{5} - 27 \beta_{3} + 6 \beta_{2} + 51$$ $$\nu^{5}$$ $$=$$ $$16 \beta_{17} - 6 \beta_{16} + 44 \beta_{15} - 27 \beta_{14} - 72 \beta_{13} - 64 \beta_{12} + 29 \beta_{6} + 12 \beta_{4} + 103 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-64 \beta_{11} - 354 \beta_{10} + 103 \beta_{9} - 215 \beta_{8} + 12 \beta_{7} - 5 \beta_{5} + 370 \beta_{3} - 154 \beta_{2} - 630$$ $$\nu^{7}$$ $$=$$ $$-215 \beta_{17} + 154 \beta_{16} - 555 \beta_{15} + 370 \beta_{14} + 1365 \beta_{13} + 1091 \beta_{12} - 613 \beta_{6} - 316 \beta_{4} - 1318 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$1091 \beta_{11} + 4741 \beta_{10} - 1318 \beta_{9} + 2863 \beta_{8} - 316 \beta_{7} - 466 \beta_{5} - 5265 \beta_{3} + 2926 \beta_{2} + 8472$$ $$\nu^{9}$$ $$=$$ $$2863 \beta_{17} - 2926 \beta_{16} + 7063 \beta_{15} - 5265 \beta_{14} - 23241 \beta_{13} - 17415 \beta_{12} + 11122 \beta_{6} + 6110 \beta_{4} + 17918 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-17415 \beta_{11} - 65509 \beta_{10} + 17918 \beta_{9} - 39149 \beta_{8} + 6110 \beta_{7} + 11654 \beta_{5} + 76701 \beta_{3} - 49692 \beta_{2} - 119143$$ $$\nu^{11}$$ $$=$$ $$-39149 \beta_{17} + 49692 \beta_{16} - 93499 \beta_{15} + 76701 \beta_{14} + 374983 \beta_{13} + 270663 \beta_{12} - 186142 \beta_{6} - 105008 \beta_{4} - 252911 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$270663 \beta_{11} + 930631 \beta_{10} - 252911 \beta_{9} + 551214 \beta_{8} - 105008 \beta_{7} - 217225 \beta_{5} - 1132827 \beta_{3} + 800346 \beta_{2} + 1720394$$ $$\nu^{13}$$ $$=$$ $$551214 \beta_{17} - 800346 \beta_{16} + 1287006 \beta_{15} - 1132827 \beta_{14} - 5879102 \beta_{13} - 4152038 \beta_{12} + 2981143 \beta_{6} + 1704192 \beta_{4} + 3657346 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-4152038 \beta_{11} - 13497390 \beta_{10} + 3657346 \beta_{9} - 7943412 \beta_{8} + 1704192 \beta_{7} + 3653944 \beta_{5} + 16867706 \beta_{3} - 12535678 \beta_{2} - 25251857$$ $$\nu^{15}$$ $$=$$ $$-7943412 \beta_{17} + 12535678 \beta_{16} - 18265698 \beta_{15} + 16867706 \beta_{14} + 90707172 \beta_{13} + 63242434 \beta_{12} - 46573888 \beta_{6} - 26818108 \beta_{4} - 53720321 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$63242434 \beta_{11} + 198554992 \beta_{10} - 53720321 \beta_{9} + 116353757 \beta_{8} - 26818108 \beta_{7} - 58566063 \beta_{5} - 252390368 \beta_{3} + 193303392 \beta_{2} + 374457783$$ $$\nu^{17}$$ $$=$$ $$116353757 \beta_{17} - 193303392 \beta_{16} + 264892277 \beta_{15} - 252390368 \beta_{14} - 1386576839 \beta_{13} - 959360641 \beta_{12} + 717246069 \beta_{6} + 414724174 \beta_{4} + 796922229 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
506.1
 0.987478i 3.88295i 3.03152i 1.03381i 1.76052i − 0.271062i − 0.199774i 2.07331i − 0.421015i 0.421015i − 2.07331i 0.199774i 0.271062i − 1.76052i − 1.03381i − 3.03152i − 3.88295i − 0.987478i
2.78942i −1.81462 −5.78084 1.00000i 5.06173i 0.269601i 10.5463i 0.292842 −2.78942
506.2 2.63597i 1.98944 −4.94835 1.00000i 5.24412i 3.28231i 7.77176i 0.957886 −2.63597
506.3 2.58648i 0.884825 −4.68989 1.00000i 2.28858i 0.858383i 6.95735i −2.21708 2.58648
506.4 2.28079i 3.21428 −3.20199 1.00000i 7.33108i 2.30369i 2.74149i 7.33158 2.28079
506.5 2.20556i −0.0130567 −2.86449 1.00000i 0.0287972i 4.60897i 1.90669i −2.99983 −2.20556
506.6 1.53088i 2.88726 −0.343581 1.00000i 4.42003i 3.86493i 2.53577i 5.33625 −1.53088
506.7 1.04721i −2.75868 0.903360 1.00000i 2.88890i 3.42366i 3.04042i 4.61031 1.04721
506.8 0.271374i −0.319618 1.92636 1.00000i 0.0867358i 3.38151i 1.06551i −2.89784 0.271374
506.9 0.0240266i 2.93017 1.99942 1.00000i 0.0704021i 1.66541i 0.0960927i 5.58589 −0.0240266
506.10 0.0240266i 2.93017 1.99942 1.00000i 0.0704021i 1.66541i 0.0960927i 5.58589 −0.0240266
506.11 0.271374i −0.319618 1.92636 1.00000i 0.0867358i 3.38151i 1.06551i −2.89784 0.271374
506.12 1.04721i −2.75868 0.903360 1.00000i 2.88890i 3.42366i 3.04042i 4.61031 1.04721
506.13 1.53088i 2.88726 −0.343581 1.00000i 4.42003i 3.86493i 2.53577i 5.33625 −1.53088
506.14 2.20556i −0.0130567 −2.86449 1.00000i 0.0287972i 4.60897i 1.90669i −2.99983 −2.20556
506.15 2.28079i 3.21428 −3.20199 1.00000i 7.33108i 2.30369i 2.74149i 7.33158 2.28079
506.16 2.58648i 0.884825 −4.68989 1.00000i 2.28858i 0.858383i 6.95735i −2.21708 2.58648
506.17 2.63597i 1.98944 −4.94835 1.00000i 5.24412i 3.28231i 7.77176i 0.957886 −2.63597
506.18 2.78942i −1.81462 −5.78084 1.00000i 5.06173i 0.269601i 10.5463i 0.292842 −2.78942
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 506.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.h 18
13.b even 2 1 inner 845.2.c.h 18
13.c even 3 2 845.2.m.j 36
13.d odd 4 1 845.2.a.n 9
13.d odd 4 1 845.2.a.o yes 9
13.e even 6 2 845.2.m.j 36
13.f odd 12 2 845.2.e.o 18
13.f odd 12 2 845.2.e.p 18
39.f even 4 1 7605.2.a.cp 9
39.f even 4 1 7605.2.a.cs 9
65.g odd 4 1 4225.2.a.bs 9
65.g odd 4 1 4225.2.a.bt 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.n 9 13.d odd 4 1
845.2.a.o yes 9 13.d odd 4 1
845.2.c.h 18 1.a even 1 1 trivial
845.2.c.h 18 13.b even 2 1 inner
845.2.e.o 18 13.f odd 12 2
845.2.e.p 18 13.f odd 12 2
845.2.m.j 36 13.c even 3 2
845.2.m.j 36 13.e even 6 2
4225.2.a.bs 9 65.g odd 4 1
4225.2.a.bt 9 65.g odd 4 1
7605.2.a.cp 9 39.f even 4 1
7605.2.a.cs 9 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 1748 T^{2} + 27292 T^{4} + 53829 T^{6} + 42488 T^{8} + 17195 T^{10} + 3912 T^{12} + 507 T^{14} + 35 T^{16} + T^{18}$$
$3$ $$( -1 - 78 T - 103 T^{2} + 385 T^{3} - 129 T^{4} - 130 T^{5} + 72 T^{6} + 3 T^{7} - 7 T^{8} + T^{9} )^{2}$$
$5$ $$( 1 + T^{2} )^{9}$$
$7$ $$361201 + 5795058 T^{2} + 11934531 T^{4} + 8104363 T^{6} + 2584363 T^{8} + 447808 T^{10} + 44736 T^{12} + 2573 T^{14} + 79 T^{16} + T^{18}$$
$11$ $$79138816 + 143149056 T^{2} + 103323904 T^{4} + 39269760 T^{6} + 8675552 T^{8} + 1156592 T^{10} + 92993 T^{12} + 4330 T^{14} + 105 T^{16} + T^{18}$$
$13$ $$T^{18}$$
$17$ $$( -6656 + 15104 T - 640 T^{2} - 10880 T^{3} - 440 T^{4} + 1652 T^{5} + 51 T^{6} - 80 T^{7} - T^{8} + T^{9} )^{2}$$
$19$ $$116985856 + 576276480 T^{2} + 804173568 T^{4} + 365179904 T^{6} + 75407600 T^{8} + 7901544 T^{10} + 432329 T^{12} + 12433 T^{14} + 178 T^{16} + T^{18}$$
$23$ $$( 43693 - 147293 T + 1626 T^{2} + 50661 T^{3} + 6823 T^{4} - 4181 T^{5} - 1081 T^{6} - 20 T^{7} + 14 T^{8} + T^{9} )^{2}$$
$29$ $$( 1 + 2485 T - 20144 T^{2} + 19533 T^{3} - 3473 T^{4} - 1963 T^{5} + 695 T^{6} - 22 T^{7} - 12 T^{8} + T^{9} )^{2}$$
$31$ $$116985856 + 1232504832 T^{2} + 3588331776 T^{4} + 3321950592 T^{6} + 552220192 T^{8} + 39342656 T^{10} + 1439097 T^{12} + 27918 T^{14} + 269 T^{16} + T^{18}$$
$37$ $$100362568044544 + 45075288696832 T^{2} + 6735685985536 T^{4} + 500587303040 T^{6} + 21358407200 T^{8} + 554796016 T^{10} + 8889217 T^{12} + 85491 T^{14} + 451 T^{16} + T^{18}$$
$41$ $$986619064369 + 1679098195957 T^{2} + 674772731916 T^{4} + 107030888677 T^{6} + 7546036571 T^{8} + 277590521 T^{10} + 5721491 T^{12} + 66394 T^{14} + 404 T^{16} + T^{18}$$
$43$ $$( 71513 - 50162 T - 96071 T^{2} - 3903 T^{3} + 36193 T^{4} + 19664 T^{5} + 4762 T^{6} + 605 T^{7} + 39 T^{8} + T^{9} )^{2}$$
$47$ $$136352329 + 9525454257 T^{2} + 15973116056 T^{4} + 7491694937 T^{6} + 1427819051 T^{8} + 116137545 T^{10} + 3677031 T^{12} + 54110 T^{14} + 376 T^{16} + T^{18}$$
$53$ $$( -4469312 + 4091904 T + 126352 T^{2} - 537968 T^{3} + 32544 T^{4} + 20856 T^{5} - 1389 T^{6} - 263 T^{7} + 8 T^{8} + T^{9} )^{2}$$
$59$ $$74472226816 + 3576704204800 T^{2} + 1105401995264 T^{4} + 139942122496 T^{6} + 9337516352 T^{8} + 353027376 T^{10} + 7577401 T^{12} + 87795 T^{14} + 491 T^{16} + T^{18}$$
$61$ $$( -183247 + 580142 T - 52953 T^{2} - 141537 T^{3} + 5939 T^{4} + 8224 T^{5} - 280 T^{6} - 159 T^{7} + 3 T^{8} + T^{9} )^{2}$$
$67$ $$13253305129 + 89951577779 T^{2} + 147684426229 T^{4} + 48274440933 T^{6} + 5013034897 T^{8} + 230544577 T^{10} + 5435904 T^{12} + 67898 T^{14} + 421 T^{16} + T^{18}$$
$71$ $$31208937558016 + 22466891294720 T^{2} + 4402169992448 T^{4} + 390994664832 T^{6} + 18739738848 T^{8} + 523076048 T^{10} + 8726265 T^{12} + 85502 T^{14} + 453 T^{16} + T^{18}$$
$73$ $$751369977856 + 1948919988224 T^{2} + 865961492480 T^{4} + 143461531648 T^{6} + 11049376768 T^{8} + 437865472 T^{10} + 9181248 T^{12} + 99072 T^{14} + 512 T^{16} + T^{18}$$
$79$ $$( -10816 - 14144 T + 129824 T^{2} - 148432 T^{3} + 43340 T^{4} + 2400 T^{5} - 3115 T^{6} + 548 T^{7} - 39 T^{8} + T^{9} )^{2}$$
$83$ $$2478502513962721 + 599300955807314 T^{2} + 58736823918955 T^{4} + 3064780382035 T^{6} + 94333579031 T^{8} + 1792122704 T^{10} + 21198204 T^{12} + 151657 T^{14} + 599 T^{16} + T^{18}$$
$89$ $$1100401 + 5490330831 T^{2} + 38614827717 T^{4} + 38681842621 T^{6} + 5353140597 T^{8} + 287501201 T^{10} + 7362892 T^{12} + 92054 T^{14} + 509 T^{16} + T^{18}$$
$97$ $$11289439903154176 + 3554298160611328 T^{2} + 424095045185536 T^{4} + 24562511884288 T^{6} + 749342566336 T^{8} + 12373327008 T^{10} + 109508329 T^{12} + 505045 T^{14} + 1142 T^{16} + T^{18}$$